Geometro-stochastic Quantization Research Articles

Geometro-differential conception of extended particles and their quantum theory in de Sitter space

A geometro-differential quantum theory of extended particles is presented. The geometrical selling is that of Hilbert fiber bundles whose base manifolds are pseudo-Riemannian space-times of points χ which are interpreted as partial aspects of physical reality (the extended particle). The fibers are carrier spaces of induced (internal configuration and momentum) representations of the structural group (the de Sitter group here). Sections of these bundles are seen as physical representations of the particle, and their values in the fibers are interpreted as states of internally localized modes composing the particle. This geometrical structure is “analogous” to that of a geometro-stochastic quantization developed in recent years. The physical interpretation is a combination of those of an old functional quantum theory and of an induced representation scheme based on an interpretation of intertwining, between configuration and momentum representations, as localization, materialization, and propagation of particles. Our model is applied in two cases: (1) Induced representation is applied in both space-time and internal space: both have de Sitter symmetry whose connection is ignored. Intertwining is considered in both spaces in a composed fashion. (2) For generic spacetimes and when the connection is taken into account, intertwining is applied only in the internal space. Parallel transport is combined with intertwining for a redefinition of localization, materialization, and propagation (the latter in a path integral context inspired from geometro-stochastic quantization).

Quantized fiber dynamics for extended elementary objects involving gravitation

The geometro-stochastic quantization of a gauge theory for extended objects based on the (4, 1)-de Sitter group is used for the description of quantized matter in interaction with gravitation. In this context a Hilbert bundle ℋ over curved space-time B is introduced, possessing the standard fiber ℋ\(_{\bar \eta }^{(\rho )} \), being a resolution kernel Hilbert space (with resolution generator\(\tilde \eta \)and generalized coherent state basis) carrying a spin-zero phase space representation of G=SO(4, 1) belonging to the principal series of unitary irreducible representations determined by the parameter ρ. The bundle ℋ, associated to the de Sitter frame bundle P(B, G), provides a geometric arena with built-in fundamental length parameter R (taken to be of the order of 10−13 cm characterizing hadron physics) yielding, in the presence of gravitation, a quantum kinematical framework for the geometro-stochastic description of spinless matter described in terms of generalized quantum mechanical wave functions, Ψxρ(ξ, ζ), defined on #x210B;. By going over to a nonlinear realization of the de Sitter group with the help of a section ξ(x) on the soldered bundle E, associated to P, with homogeneous fiber V′4⋍G/H, one is able to recover gravitation in a de Sitter gauge invariant manner as a gauge theory related to the Lorentz subgroup H of G. ξ(x) plays the dual role of a symmetry-reducing and an extension field. After introducing covariant bilinear source currents in the fields Ψxρ(ξ, ζ) and their adjoints determined by G-invariant integration over the local fibers in ℋ, a quantum fiber dynamical (QFD) framework is set up for the dynamics at small distances in B determining the geometric quantities beyond the classical metric of Einstein's theory through a set of current-curvature field equations representing the source equations for axial vector torsion and the de Sitter boost contributions to the bundle connection (the latter defining the soldering forms of the Cartan connection in P(B, G) in the nonlinear gauge). The presented bundle framework yields a theory for quantized material objects in interaction with gravitation, the long-range metrical part of which remains classical.

Geometro-stochastic quantization of a theory for extended elementary objects

The geometro-stochastic quantization of a gauge theory based on the (4,1)-de Sitter group is presented. The theory contains an intrinsic elementary length parameter R of geometric origin taken to be of a size typical for hadron physics. Use is made of a soldered Hilbert bundle ℋ over curved spacetime carrying a phase space representation of SO(4, 1) with the Lorentz subgroup related to a vierbein formulation of gravitation. The typical fiber of ℋ is a resolution kernel Hilbert space ℋ $$_{\bar \eta }^{(\rho )} $$ constructed in terms of generalized coherent states $$\bar \eta $$ ρ related to the principal series of unitary irreducible representations of SO(4, 1), namely de Sitter horospherical waves for spinless particles characterized by the parameter ρ. The framework is, finally, extended to a quantum field-theoretical formalism by using bundles with Fock space fibers constructed from ℋ $$_{\bar \eta }^{(\rho )} $$ .

Geometro-stochastic locality in quantum spacetime and quantum diffusions

The issue of the intrinsic nonlocality of quantum mechanics raised by J. S. Bell is examined from the point of view of the recently developed method of geometro-stochastic quantization and its applications to general relativistic quantum theory. This analysis reveals that a distinction should be made between the topological concept of locality used in formulating relativistic causality and a type of geometric locality based on the concept of fiber bundle, which can be used in extending the strong equivalence principle to the quantum domain. Both play an essential role in formulating a notion of geometro-stochastic propagation based on quantum diffusions, which throws new light on the EPR paradox, on the origin of the arrow of time, and on other fundamental issues in quantum cosmology and the theory of measurement.

Foundational aspects of geometro-stochastic quantization of Dirac fields in curved spacetime

The application of the recently developed method of geometro-stochastic quantization to spin-1/2 fields in curved spacetime does not give rise to any of the conceptual and technical difficulties encountered by more conventional methods of quantization. Field propagation is based on the concept of stochastic parallel transport governed by quantum connections that incorporate Poincare gauge invariance and obey the equivalence principle. Consequently, under free-fall conditions, such geometro-stochastic Dirac field propagation does not give rise to Bogolubov transformations resulting in spontaneous fermion creation.

Geometro-stochastic quantization of gauge fields in curved space-time

Geometro-stochastic quantization of gauge fields in curved space-time

Geometro-stochastic quantization of gravity. II

Geometro-stochastic quantization of gravity. II

Geometro-stochastic quantization of gravity. I

A general method of geometro-stochastic first quantization is outlined, and then applied to the construction of a graviton bundleE. This quantization is performed by treating the graviton as a massless stringlike exciton of spin 2, whose states can be represented by the vectors in the bundleE over a base manifoldM of mean spacetime locations. The graviton bundleE carries an indefinite inner product, and possesses both external Poincare gauge degrees of freedom, as well as internal gauge degrees of freedom related to general changes of coordinates in an immediate neighbourhood of eachx ∈M.

Geometro-stochastic quantization of gauge fields in curved space-time

It is shown that the geometro-stochastic method of quantization of massive fields in curved space-time can be extended to the massless cases of electromagnetic fields and general Yang-Mills fields. The Fock fibres of the massive case are replaced in the present context by fibres with indefinite inner products, such as Gupta-Bleuler fibres in the electromagnetic case. The quantum space-time form factor used in the massive case gives rise in the present case to quantum gauge frames whose elements are generalized coherent states corresponding to pseudounitary spin-one representations of direct products of the Poincare group with theU(1),SU(N) or other internal gauge groups. Quantum connections are introduced on bundles of second-quantized frames, and the corresponding parallel transport is expressed in terms of path integrals for quantum frame propagators. In the Yang-Mills case, these path integrals make use of Faddeev-Popov quantum frames. It is shown, however, that in the present framework the ghost fields that give rise to these frames possess a geometric interpretation related to the presence of a super-gauge group that, in addition to the external Poincare and Yang-Mills gauge degrees of freedom, involves also the internal ones related to choices of gauge bases within the quantum fibres.

On stringlike excitons in the geometro-stochastic quantization on gravity

A connection between string theory and the geometro-stochastic method of quantizing gravity is derived by applying Born's concept of quantum metric operator to massless exciton states. The graviton is the identified with the spin-2 ground exciton state of the relativistic harmonic oscillator resulting from combining the string equation with the equations of Born and Yukawa.

Geometro-stochastic quantization of massive fields in curved space-time

It is shown that an earlier introduced concept of quantum geometry, given by a Hilbert bundle over a base space whose elements constitute a generic curved space-time and represent the mean locations of quantum frames, can be generalized to the case of Fock bundles, whose Fock fibres are the carriers of multiparticle states. In this new context, the role of quantum Lorentz frame is taken over by second-quantized frames constructed out of coherent exciton states. The structure groups of the considered bundles contain unitary representations of the Poincare group, with the latter emerging as a gauge group for the internal degrees of freedom of the considered quantum particles. Connections compatible with the Hermitian structure are introduced on bundles of second-quantized frames that are associated to principal bundles of affine Lorentz frames. The corresponding parallel transport is expressed in terms of path integrals for quantum frame propagators. It is shown that the resulting geometro-stochastic quantum field theory in curved space-time does not give rise to the foundational difficulties with the particle concept, with normal ordering and with the definition of the stress-energy tensor, that are inherent in more conventional approaches to quantum field theory in curved space-time. Hence, the derived path integral formulae for the propagation of systems of quantum field particles display in the present framework none of the ambiguities encountered by those approaches.